# Four-term progression free sets with three-term progressions in all large subsets

## TL;DR

This paper constructs large sets without four-term arithmetic progressions that still contain three-term progressions in all large subsets, advancing understanding of progression-rich sets in finite fields and integers.

## Contribution

It introduces new constructions of progression-free sets with rich three-term progressions in large subsets, and improves quantitative Roth-type theorems in random subsets.

## Key findings

- Existence of four-term progression-free sets with all large subsets containing three-term progressions
- Improved bounds in Roth-type theorems for random subsets of finite fields
- Demonstration that three-term progression richness is nearly uncorrelated with additive energy

## Abstract

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three term arithmetic progressions, in the sense that all sufficiently large subsets contain at least one such progression. We prove that there exists a positive constant $c$ and a set $A \subset \mathbb F_q^n$ which does not contain a four-term arithmetic progression, with the property that for every subset $A' \subset A$ with $|A'| \geq |A|^{1-c}$, $A'$ contains a nontrivial three term arithmetic progression. We derive this from a more general quantitative Roth-type theorem in random subsets of $\mathbb{F}_{q}^{n}$, which improves a result of Kohayakawa-Luczak-R\"odl/Tao-Vu. We also discuss a similar phenomenon over the integers, where we show that for all $\epsilon >0$, and all sufficiently large $N \in \mathbb N$, there exists a four-term progression-free set $A$ of size $N$ with the property that for every subset $A' \subset A$ with $|A'| \gg \frac{1}{(\log N)^{1-\epsilon}} \cdot N$ contains a nontrivial three term arithmetic progression. Finally, we include another application of our methods, showing that for sets in $\mathbb{F}_{q}^{n}$ or $\mathbb{Z}$ the property of "having nontrivial three-term progressions in all large subsets" is almost entirely uncorrelated with the property of "having large additive energy".